To solve the problem using a factorial method, let's break it down step by step:

Step 1: Total Number of 5-Digit Numbers

We are forming 5-digit numbers using the digits $0, 1, 2, 3, 4$.

The first digit (the most significant digit) cannot be $0$, while the remaining digits can be any of $0, 1, 2, 3, 4$.

• For the first digit, we have 4 choices: $1, 2, 3, 4$.
• For the remaining four digits, each digit has 5 choices: $0, 1, 2, 3, 4$.

The total number of possible 5-digit numbers is: $4 \times 5^4 = 4 \times (5 \times 5 \times 5 \times 5) = 4 \times 625 = 2500$

Step 2: Counting the Favorable Outcomes

We need to find the number of 5-digit numbers where the digit $2$ is in the tens place and the digit $0$ is in the units place.

• The tens place is fixed as $2$.
• The units place is fixed as $0$.
• The first digit (thousands place) has 4 choices: $1, 2, 3, 4$.
• The second digit has 5 choices: $0, 1, 2, 3, 4$.
• The third digit has 5 choices: $0, 1, 2, 3, 4$.

The number of favorable outcomes is calculated by multiplying these choices together: $4 \times 5 \times 5 = 100$

Step 3: Calculate the Probability

The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes: $\text{Probability} = \frac{100}{2500} = \frac{1}{25}$

Final Answer

Using the factorial method, the probability of getting $2$ in the tens place and $0$ in the units place is $\boxed{\frac{1}{25}}$.

Would you like to delve deeper into the factorial method or have any questions?

Here are some related questions:

1. How would the probability change if the digits were allowed to repeat?
2. What is the probability of the first digit being $3$ and the last digit being $1$?
3. Can you calculate the probability using permutations instead of combinations?
4. What is the probability of all digits being odd numbers?
5. How many 5-digit numbers can be formed if each digit must be different?

Tip: Factorials are very useful for calculating probabilities, especially when dealing with permutations and combinations in probability theory.